.=method is a mutating call of the method on an object. The call $x.=method does the same as the more verbose assignment $x = $x.method.
In the following example, the $o container initially holds an object of the C class, but after $o.=m(), the value will be replaced with an instance of the D class.
class D { }Â
class C {
   method m() {
       return D.new;
   }
}
my $o = C.new;
say $o.WHAT;Â # (C)Â
$o.=m();
say $o.WHAT;Â # (D)